Earn free rewards

Help us pay bills
(no cost to you)
use these links to
visit advertisers

the Five-Room puzzle

If you have not yet seen my main information page about the Five Room puzzle, I suggest that you click here and read it first. This page builds on the information from that page and goes into much more detail.

The 5-room puzzle (a collection at www.MAZES.com):

On the first information page: The impossible 5-room puzzle Two variations that can be solved Requirements for solvability Why the puzzle is impossible Cheating in Two Dimensions Cheating in Three Dimensions On this page: Introduction The puzzle (impossible on paper) Puzzle Variations that are also impossible Why it's impossible in 2 dimensions The Bridges of Konigsberg Puzzle (impossible) How do you Cheat

The Five Room Puzzle
Why it cannot be solved
and
How to cheat to get an answer

Introduction

This page regards the "Five Room Puzzle" that can be found at

www.mazes.com/fiverooms.html
(or if that link doesn't work, at this secondary address:
http://www.MAZES.com/fiverooms.html

If you are not acquainted with that puzzle, I suggest you read the original puzzle page (above) before you read this page, which is a reply to a student who was exasperated with that puzzle. You might want to attempt to solve it before you read my explanation and other versions of the puzzle, below.

Suggestion - keep reading even if you get confused

This quick introductory note is to let you know that there will be some things in this reply that you might not understand. If that happens, please keep reading, because I will actually explain this puzzle with several variations. So, even if you don't understand one section, you might understand the next, then you might come back to the confusion section and understand it better the second time.

An e-mail from a perplexed puzzler

A student wrote to me to tell me that my

"5 room puzzle is driving (her) crazy!"

She went on to tell me that she had

"probably drawn (my) little 'house' hundreds of times"

and she

"STILL can't figure out the solution. What is it?!?"

Here is the e-mail that I wrote to her, modified slightly for web publication, including a review of the original puzzle.

My reply to Megan about
The Five Rooms Puzzle

The reason that you can't find a solution is because you are doing it on paper. This puzzle cannot be solved in two dimensions (in other words, it cannot be solved on one side of a flat surface).

You might stop here and try to solve the puzzle again, because the paragraph above is a big HINT about what you might have to do in order to solve the puzzle. More on that further below. (And to add to that hint, it also cannot be solved on the surface of a sphere, or anything topologically equivalent to a sphere. Don't worry if you don't know what topology is, many students don't learn about topology until high school or college. I learned about it in my senior year of high school, I think.)

Let's come back to regular 2-D space for awhile, and ...

Create the first version of the puzzle on Paper:

Let me try to explain why it can't be done by having you create the puzzle on paper and looking at it.

Let's think about the house as a diagram with six rooms. What? You say, the house only has five rooms?

The outside of the house counts as one room, the walls of which are the four walls of the house. If you were to draw the house on a very-elastic balloon, you could stretch and shrink parts of the balloon and turn it into a room inside the house with one of the other rooms on the outside, but that's something that's hard to explain, other that to say that you will learn about when your teacher tells you about "topology". (How to see this sixth room: 1) Draw the diagram in such a way that the walls of the house are at the very edges of the paper. 2) Fold over the very edges so only the walls are folded around to the back of the paper. 3) Look at the back of the paper and you see a room that has nine doors.)

Here's what I want you to do:

1. Draw a picture of the house on paper

2. Put a dot in the middle of each room, and put one dot somewhere outside the house.

3. Draw lines from each dot to each door and put a number on (or next to) the dot counting those lines

As you can see, I ended up with 5 lines coming out of each dot in three of the rooms, 4 lines coming out of the dot in two of the rooms, and 9 lines coming out of the dot that is "inside" the "outside room" of the diagram. In other words, four of the numbers are odd.

In essence, we have two related puzzles.

1. the original "start inside any room and go through every doorway exactly once" and
2. or a revised "start on any dot and follow every line exactly once"

In order to prove to you why we cannot solve the 5 room puzzle, I am going to use the second version of the puzzle which, with numbers, can be somewhat mathematical.

Variations that might make the impossibility easier to see

The second puzzle, with slightly more involved instructions:
I've (almost) erased the walls so that we can emphasize the paths.

1. Start on any dot
2. How to follow a line from that dot to the next dot:
   • subtract one from the number on that dot and write down the new number there (for example, change 5 to 4)
   • erase the line leading from that dot to the next dot (or just cross out the line so you know you used it)
   • subtract one from the number on that next dot and write down the new number
     (the new numbers represent how many lines are still available to be used touching that vertex)
3. Keep repeating step 2 until the new number is zero (because all paths are now erased, and you can't leave the dot, you're stranded.

You will probably quickly notice that no matter what happens, you will eventually end up at a point with no paths left, and still have somewhere else that you cannot get to. One of the times I tried to do this, I started in the top right room and ended up in the top left room (where the arrow is pointing here), and I had no way to get to the last line that was between the two remaining 1's. (If I had started on one of the 4's, I would have ended up with two separate lines that I couldn't get to.

The third puzzle only has numbers:

Now, let's look at it as another kind of puzzle, one in which we ONLY look at the numbers.

This puzzle has fairly simple instructions. First, let's set it up:

1. Put five pennies on top of each of the three circles with 5 on them
2. Put four pennies on top of each of the two circles that have the 4 on them
3. Put nine pennies on top of the 9 circle

Now, here are the directions for this third puzzle:

1. Start on any pile of coins by putting your finger there.
2. We are going to jump from pile to pile, but each time we jump, we are going to remove two pennies.
   • Remove a penny when you move away from the pile you are leaving
   • Remove a penny when you arrive on the new pile
3. Keep jumping from pile to pile until the pile you jump to has only one penny on it (which you take away when you land there.

Or here's an even easier set of directions to do the same thing:

1. Start on any pile. Remove one penny from that pile.
2. Jump to any other pile that has two or more pennies on it and remove two pennies from that pile.
   Keep repeating step two until there are no piles with two or more pennies on it.
3. Finally, when there are no more larger piles left, jump to any pile that has just one penny on it, remove that penny and STOP. The game is over.
   If you started on an odd pile, you ended up on another odd pile. If you started on an even pile, I don't know where you ended up. You could have ended up back where you started, or on one of the odd piles.

Even though this puzzle is much simpler, we can still see why it is impossible to solve, because we always have to remove an even number of pennies for every step except for the first and last steps. Since there are four odd numbers, we can never get to all zeroes, because we can only subtract 1 twice. Two of the 1's will always be left.

Why is this puzzle impossible?

By now, you know that this puzzle is impossible to solve in two dimensions, whether you do it as doors in a house, lines on a diagram, or pennies in piles.

The Mathematician Euler was the one who created a line diagram to prove that a different (but similar) puzzle known as the Bridges of Konigsberg could not be solved. Our line diagram for the second version of the puzzle (above) is similar to the diagram that he used to prove that the Bridges puzzle was unsolvable.

He proved that this kind of puzzle can ONLY be solved if it has either zero or two odd vertexes. (Can you look at that diagram and tell me why you could never have one odd vertex? If you're not sure, try to draw a diagram that has an odd number of odd vertices. I bet you can't do it). Our puzzle has four odd vertexes (your math teacher might prefer you to use the word vertices?). If a line diagram has zero odd vertices, you can start on any vertex and when you finish, you will end up back at that same vertex. If you have two odd vertexes, you must start on one of the odd vertexes and you will end on the other odd vertex (if you want to solve it, that is). Part of his proof was based on the observation was that each time you went through a room, you reduced the number of doors in that room by two.

In fact, that gives me an idea to illustrate the numbers example. Let's imagine the first puzzle again. Start with all the doors open. As you go through a door, close it behind you. With this example, each time you go through a room, you close two doors, one as you enter the room and one as you leave the room. Eventually, you enter a room, close the door behind you, and find yourself in a room with all doors closed. In this case, the piles of pennies illustrate open doors in the room.

The Bridges of Konigsberg

There is another variation of this puzzle that Euler used to prove that this type of puzzle. It was an actual puzzle that was attempted by boyfriends and girlfriends in Russia, in a town with two islands in a river that divided the town. Let's look at a simple map of that town:

The object of this variation of the puzzle is to start anywhere in the town and cross every bridge exactly once (without flying or swimming). In other words, you can only walk around this town, which was known as Konigsberg, which is why this puzzle is called the "Bridges of Konigsberg" puzzle.

If you wanted to turn this into a Euler-style diagram, put a vertex on each side of the river and one on each island, and count the number of bridges that reach that piece of land. You will have a three on each of the four vertexes, and if you remember, Euler proved that you can only solve a line puzzle like this if there are no more than two odd numbers. Here is one way to draw those vertices:

No matter what you do, we cannot start anywhere and cross every bridge. As Euler explained, we have to start on an odd number, then each time we cross two bridges, we have reduced the number between those two bridges by two.

Let's review, again, the reason you can't solve this variation of the puzzle. We start at any number and reduce it by one. Then we reduce every succeeding number by two (as we cross a bridge to that piece of land then leave that piece of land again, using two bridges in the process). Finally, when we get to a piece of land that only has a one on it, we reduce that number to zero and stop, because we can't subtract two from one (no negative numbers in this puzzle), and there are two other 1's that can't be reached. Basically, the only numbers that permanently change to even are the place where you start and the place where you finish. All the other odd numbers simply change from 3 to 1 or from 5 to 3 to 1.

Back to that number variation of the puzzle:

Or to summarize that number variation we had.

1. Start on any odd number, subtract 1 from it to make it even
2. Go to any number greater than 1 and subtract 2 from it
3. Keep repeating step 2 until there are no numbers greater than 1 left
4. Go to one of the remaining 1's and subtract 1 from it, leaving zero (and QUIT)

You can do this with any number of pennies (or bean counters or whatever).

1. Start with any numbers of piles of counters.
2. Put any number of counters in each pile (make sure there are an even number of odd numbers, if that makes sense)
3. Write down the starting number for each pile
4. Follow these rules for the game (similar to above)
   1. Take one counter away from your starting pile.
   2. Keep taking two counters away from any pile that has two or more counters in it until there are no piles with two or more counters on them
   3. Take one counter away from your ending pile.
5. If you start with more than two odd piles, you will end with some untouched counters.
6. If you started with two odd piles, and if you started by removing one counter from an odd pile, you solved the puzzle, but if you started on an even pile, you ended up with two counters left at the end.

How do you cheat on these puzzles?

OK ... I've proved to you why the puzzle is impossible to solve, so how can you solve it? As I indicated above, you have to think three dimensionally.

Let's look at the bridges of Konigsberg puzzle again. We could cheat with this puzzle by building a "Chunnel" ... which is the name that some people call that tunnel between France and England that goes under the English Channel. If you add a "chunnel" to the city of Konigsberg, you are changing the numbers on the shores of the river to 4's, because 3 bridges and the chunnel come out on the shore. This underground bridge reduces the puzzle to a diagram with two odd numbers, a solvable puzzle if you start on one island and end on the other. The puzzle is still not solveable if you start on either shore because those are now even numbers.

The thin lines show the approximate route of the underground tunnel between the two sides of the river. That is one way to cheat with the Bridges of Konigsberg puzzle. You could also solve this puzzle by drawing it onto a cylinder (such as the core of a roll of paper towels). Drawing the diagram onto a cylinder allows you to continue the line from one shore around the tube to the other shore, another way of going UNDER the diagram.

Can you think of a similar way to cheat with the five room puzzle?

A hint for cheating with the five-room puzzle:

Think three dimensionally. Just like we used the third dimension on the bridges puzzle (we went UNDER the river), can you use the third dimension in the house puzzle (like going under the house? or simpler would be to go UNDER the first floor of the house?)

With that much of a hint, you should be able to cheat and build an answer to the puzzle. Just build a house with a basement underneath.

See that middle room in the one row that has five doors in it? Let's call it the kitchen and put a trap door in the floor that goes into the basement. You know the type, there's a rope in the piece of wood, you pull up on it, and you jump down through the hole in the floor. That takes care of an extra door in the kitchen. It's a door in the floor.

But how do we get out of the basement?

Have you seen the old Judy Garland movie "Wizard of Oz"? I bet you have. Remember the doors on the ground that lifted out so they could get into the basement. Let's put one of those on the outside.

What have we done? The kitchen now has six doors, counting the trap door under the rug as a door, and the outside now has ten doors, the way to escape from the basement. But we've added a seventh room to the puzzle, a basement that has two doors, neither of which are on the first floor (though one is in the first floor, in the kitchen floor). Now you have a solveable puzzle. Just start in one of the rooms that still has 5 rooms, and you can solve the puzzle. (You will end up in the other room that still has 5 doors.)

BUT ... your teacher probably won't let you do that, so let's come up with a way of cheating that your teacher almost has to accept (if you were doing it as a class assignment).

A more elegant way to cheat?

Can you think of something that some people keep on their counter (something to eat, sort of bready like, hard on the outside, soft on the inside) that you could draw the five room puzzle on and be able to solve the puzzle? YEP ... try drawing the puzzle on a misshapen BAGEL (one where the hole is closer to one side than the other). Draw it such that the hole in the bagel is in the middle of the middle room (the kitchen) that has five doors in it. (You could draw it onto a doughnut, but the icing might make it hard to write clearly.)

Basically, by the rules of topology, the inside of the kitchen is also outside the house, because you can get from the inside of the kitchen to the outside of the house without crossing any lines, by going through the hole, drawing your line around the back side of the bagel back around to the front.

Now, I bet you can solve the puzzle. At some point, you will draw a line into that kitchen room and keep drawing the line down through the hole of the bagel out to the outside of the bagel. Eventually, you should be able to finish and go through every door. Hope this helps a bit. And if you want to know more about topology, ask your math teacher to show you a mobius strip and a mobius bottle. A mobius strip is a piece of paper that only has one side. A mobius bottle is a container that has only one surface (it can only be imagined in three dimensions).

This article was prepared by John Knoderer, the webmaster at www.MAZES.com, the founder of www.GodLovesEveryone.org. If you would like to reach him, just assemble his e-mail address from the word "webmaster", the required @ symbol, and the name of one of his domains, without the www.

And if you appreciated the help that you got from this page, we urge you to make a gift to one of our web organizations. (We are setting up a non-profit organization, so eventually, gifts will be tax-deductible). You can use PayPal to send an electronic check by using this link:

Send an electronic check by using the button above

or send a credit card donation by using the button below

PayPal addresses for:

Credit Card donations: CreditCards (at) GodLovesEveryone.org
Electronic Checks: ElectronicChecksOnlyThruPayPal (at) GodLovesEveryone.org

copyright 2004 by the author.

If you enjoy puzzles, please visit <www.MAZES.com>. If you'd like to thank me for the time that went into creating this page full of information (many days of effort), please take advantage of our links. For example, if you are going to purchase some books from Amazon.com, click our links first. We will prepare other pages of information about ways that you can help us. Check back and see what else you can do (and feel free to make donations).

John
Webmaster (at) Mazes.com
www.MAZES.com

P.S. People who enjoy turning phone numbers into words might be interested in knowing that our area code spells GRY, certainly an appropriate area code for the webmaster of www.MAZES.com, a puzzle page. Unfortunately, I couldn't get MAZEMAN as a phone number (because our phones don't have Z's on them), but I did get BYTEMAN. So, I guess that I can call myself "1-GRY-BYTEMAN".

The GRY collection at www.MAZES.com:
Our webmaster's thorough explanation One Hundred Words that end in -gry: Another GRY expert and his explanation A third expert's GRY point of view	www.MAZES.com/gry.htm (this page) www.MAZES.com/gry-2.htm www.MAZES.com/gry-3.htm http://www.word-detective.com/gry.html

Copyright 1998/2004 by Rev. John Amazing, D.Div.
<MazeManJohn(nospam)(at)(nospam)Mazes.com>

This page was originally created in 1998, and updated in 1999 & 2004.

If you enjoy what you learn from any of my articles, I invite you to visit my web pages:

If you really appreciate the effort that went into the development of John's many pages,
we encourage you to send appropriate donations to the author:

The "Amazing" Rev. John

www.GodLovesEveryone.org
P O Box 235
Sulphur Springs, ARkansas 72768-0235 USA
Telephone number: 1-GRY-BYTEMAN (1-479-298-3626)
(GRY = a famous puzzle)

***** end of article *****